Classification of $T^2$-bundles over $T^2$

The geometry of symplectic pairs

“…This can be seen as generalizing a result of Geiges [8], who proved that these manifolds admit symplectic structures. His proof, like ours, depends in the classification of T 2 -bundles over T 2 due to Sakamoto and Fukuhara [22], and on the fact that all these manifolds carry compatible Thurston geometries; cf. [25].…”

“…Appendix: Orientable T 2 -bundles over T 2 Table 1 summarizes the classification of orientable T 2 -bundles over T 2 due to Sakamoto and Fukuhara [22], and the information about their Thurston geometries due to Ue [25]; compare also [8].…”

Almost toric symplectic four-manifolds

“…It is straightforward, except for the Enriques surface, to deduce from the existing literature ( [32], [9], [23]) that each of the manifolds listed in Table 1 does indeed admit an almost toric fibration. (While one would certainly expect the Z 2 quotient of the K3 to be the Enriques surface, this requires proof since the base is RP 2 rather than CP 1 as in the holomorphic case; see Lemma 5.12).…”

“…Referring then to [23] we see that the torus bundles specified by the integers (λ, m, n) and (λ ′ , m ′ , n ′ ) are equivalent if and only if λ ′ = ǫλ and n ′ = ǫn where ǫ ∈ {1, −1}, and m ′ = m + kλ + ln for some integers k, l. Furthermore, if λ = 0 or n = 0 (or equivalently, if b 1 ≥ 3), then the total space is diffeomorphic to the total space of a fibration determined by (µ, 0, 0) for some µ; otherwise the diffeomorphism classification agrees with the bundle classification. Example 5.8.…”

The transverse structure of Lie flows of codimension 3

“…Using now the classification of such bundles given by Sakamoto-Fukuhara [SF83] we obtain that M is diffeomorphic to (T 2 × R × R)/ ∼ , the quotient space of (T 2 × R × R) by the equivalence relation " ∼ " generated by (π(s, t), x, y) ∼ (π(s, t), x + 1, y) and (π(s, t), x, y) ∼ (π(A(s, t) + x(m, n)), x, y + 1)…”